On the number of good approximations of algebraic numbers by algebraic numbers of bounded degree

LXXXIX.2 (1999)

On the number of good approximations of

algebraic numbers by algebraic numbers of bounded degree

by

Helmut Locher (Marburg)

1. Introduction. Let α be an algebraic number. Roth’s celebrated the- orem [13] says that for any δ > 0 there are only a finite number of rational approximations x/y of α with

(1.1) |α − x/y| < 1/y

, y > 0.

In this paper we consider approximations of α by algebraic numbers of bounded degree. More precisely, let d ∈ N and suppose µ > 2. We look for solutions in algebraic numbers β of degree ≤ d of the inequality

(1.2) |α − β| < H

(β)

,

where H

(β) denotes the maximum modulus of the coefficients of the mini- mal defining polynomial of α over Z. For rational β, say β = x/y, we have H

(β) = max{|x|, |y|} and hence for d = 1 the inequality (1.2) is essentially equivalent to (1.1).

Wirsing [19] proved that (1.2) has for

(1.3) µ > 2d

only a finite number of solutions.

As a consequence of his famous subspace theorem W. M. Schmidt [15]

was able to prove the best possible result ([16], p. 278): (1.2) has for

(1.4) µ > d + 1

only a finite number of solutions.

Unfortunately, the underlying method of Thue–Siegel–Roth is ineffective in the sense that it does not provide upper bounds for y or H

(β) respec- tively. However, it allows giving an explicit upper bound for the number of x/y ∈ Q satisfying (1.1). A first result was proved by Davenport and Roth ([3], 1955). This bound was improved by Bombieri and van der Poorten ([1], 1987) and independently by Luckhardt ([10], 1989) using the modified proof

1991 Mathematics Subject Classification: Primary 11J68.

[97]

of Roth’s Theorem presented by Esnault and Viehweg ([4], 1984). The latest results are due to Evertse ([7], 1996, [8], 1998).

It is the purpose of this paper to prove such a quantitative result of Wirsing’s theorem.

To state our theorems we have to define the height of an algebraic num- ber. Let K be a number field and M (K) its set of places. For v ∈ M (K) denote by | · |

the associated absolute value, normalized so that on Q we have | · |

= | · | (standard absolute value) if v is archimedean, whereas for v non-archimedean |p|

= p

if v lies above the rational prime p. We put

k · k

= | · |

,

where K

denotes the completion of (K, |·|

) and Q

denotes the completion of (Q, | · |

). We also denote the unique extensions of | · |

and k · k

to K

by | · |

and k · k

respectively. For x ∈ K we define the height of x by

(1.5) H(x) = Y

v∈M (K)

max{1, kxk

}.

Let | · | denote the standard absolute value of the complex numbers C, and Q the algebraic closure of Q in C. For any positive number x we define log

x = log x if x ≥ e and 1 otherwise.

The following is a quantitative version of the result (1.3) of Wirsing ([19], Theorem 1).

Theorem 1. Let 0 < δ ≤ 1, d ∈ N and α be an algebraic number of degree f . Consider the inequality

(1.6) |α − β| < H(β)

to be solved in elements β ∈ Q with

(1.7) deg β ≤ d.

(i) There are at most

e

· d

log(6f )

δ

log d log(6f ) δ

solutions β ∈ Q of (1.6) and (1.7) with H(β) ≥ max{4

, H(α)}.

(ii) There are at most

log

log H(α)

log(1 + δ/(4d

)) + 2

δ

solutions β ∈ Q of (1.6) and (1.7) with H(β) < max{4

, H(α)}.

We suppose every number field to be embedded in Q and every valuation

of the number field to be extended to Q. The following generalizes Theorem

1 to include non-archimedean primes.

Theorem 2. Let 0 < δ ≤ 1, d ∈ N and F/K be an extension of number fields of degree f . Let S be a finite set of places of K of cardinality s.

Suppose that for each v ∈ S we are given a fixed element α

∈ F . Let H be a real number with H ≥ H(α

) for all v ∈ S. Consider the inequality

(1.8) Y

v∈S

min{1, kα

− βk

} < H(β)

to be solved in elements β ∈ Q with

(1.9) [K(β) : K] ≤ d.

Then there are at most

e

· d

log(6f )

δ

log d log(6f ) δ solutions β ∈ Q of (1.8) and (1.9) with

(1.10) H(β) ≥ max{H, 4

}.

We have claimed above that Theorems 1 and 2 are quantitative versions of Wirsing’s result (1.3). But in our theorems we have the exponent 2d

instead of 2d. The reason is that our height H(·) as defined in (1.5) is normalized in a different way than the height H

(·) in (1.2). For algebraic numbers β of degree ≤ d we have ([17], Chapter I, Lemma 7B)

H(β)

H

(β) 

H(β)

.

Therefore we get an additional factor d in the exponent. For the height H(·) the best possible exponent in (1.6) and (1.8) would be d(d + 1).

To prove the best possible result Schmidt uses an induction argument which depends upon his subspace theorem. It is not clear how this argument can be used to obtain a quantitative result.

Independently, J.-H. Evertse [8] also proved a quantitative version of Wirsing’s theorem (1.3). Moreover, he gave an explicit upper bound for the number of solutions of a more general problem considered by Wirsing [19].

His upper bounds for (1.3) are similar to ours.

2. The auxiliary polynomial

2.1. A generalization of the index. Let P be a non-zero polynomial in m variables X

, . . . , X

with complex coefficients. Roth [13] introduced the index of a polynomial at a certain point to measure to what extent the polynomial vanishes at that point. In this section we will define a different measure for this need. It was introduced by W. M. Schmidt ([18], p. 139).

Let α ∈ C

and r ∈ N

. We write P in the form P (X

, . . . , X

) = X

i

a

(α)(X

− α

)

. . . (X

− α

)

with i = (i

, . . . , i

) and unique coefficients a

(α). Let M be a subset of R

and put

k

(P ) = {(i

/r

, . . . , i

/r

) : a

(α) 6= 0}.

We say P is M-centered at α with respect to r if k

(P ) ⊆ M .

2.2. Estimation of volumes. Suppose 0 ≤ γ ≤ 1 and ε > 0. Let m ∈ N.

We put

ξ

(x) = |{h ∈ {1, . . . , m} : 0 ≤ x

≤ γ}| = X

χ

(x

),

where χ

denotes the characteristic function of the closed interval [0, γ].

The sets

M

(m, γ) = {x ∈ [0, 1]

: ξ

(x) ≤ m(γ + ε)}, M

(m) = \

γ∈[0,1]

M

(m, γ) are the main objects of this section. We always consider the complement of M

(m, γ) and M

(m) with respect to [0, 1]

. More precisely, we put

M

(m, γ) = [0, 1]

− M

(m, γ) and M

(m) = [0, 1]

− M

(m).

Lemma 2.1. Suppose 0 ≤ γ ≤ 1, ε > 0 and let m ∈ N. Then

(2.1) \

M_ε^c(m,γ)

dx ≤ e

.

The line of the proof is the same as the proof of [16], Chapter V, Lemma 4C.

P r o o f. The integral on the left-hand side of (2.1) exists, since the bound- ary of M

(m, γ) lies in a finite union of hyperplanes. For all x ∈ M

(m, γ) we have ξ

(x) − mγ > mε. Therefore

(2.2) e

\

M_ε^c(m,γ)

dx

≤ \

M_ε^c(m,γ)

e

dx = \

M_ε^c(m,γ)

e

dx

= \

M_ε^c(m,γ)

e

dx ≤ \

[0,1]^m

Y

e

dx

=



\

0

e

dx



.

Note that e

≤ 1 + y + y

for |y| ≤ 1. Hence we get (2.3)

1

\

0

e

dx

≤

1

\

0

(1 + γε(χ

(x) − γ) + γ

ε

(χ

(x) − γ)

) dx

≤ 1 + γε

1

\

0

(χ

(x) − γ) dx + γ

ε

= 1 + γ

ε

. (2.2) and (2.3) together give

e

\

M_ε^c(m,γ)

dx ≤ (1 + γ

ε

)

≤ e

and the lemma follows.

Lemma 2.2. Suppose 0 < ε ≤ 2/3 and let m ∈ N. Then

\

M_ε^c(m)

dx < 2e

.

P r o o f. In analogy to Lemma 2.1 the integral on the left-hand side exists since the boundary lies in a finite union of hyperplanes. We put

n = (

ε

(1 − ε) if

(1 − ε) ∈ N,



ε

(1 − ε) 

+ 1 otherwise;

γ

= i · ε 2



1 ≤ i ≤

 2

ε (1 − ε)



, γ

= 1 − ε.

For every γ ∈ [0, 1 − ε] there exists an i ∈ {1, . . . , n} with

(2.4) γ

− ε/2 ≤ γ ≤ γ

.

Next we show that

(2.5) M

(m) ⊇

\

M

(m, γ

).

Trivially, ξ

(x) ≤ m, and so we have M

(m) = \

γ∈[0,1]

M

(m, γ) = \

γ∈[0,1−ε]

M

(m, γ).

Now let γ ∈ [0, 1 − ε]. Take i ∈ {1, . . . , n} satisfying (2.4). Since ξ

(x) is non-decreasing in γ, for all x ∈ T

j=1

M

(m, γ

) we get

ξ

(x) ≤ ξ

(x) ≤ m(γ

+ ε/2) ≤ m(γ + ε/2 + ε/2) = m(γ + ε)

and hence x ∈ M

(m, γ). Thus we have verified (2.5).

From (2.5) we get by De Morgan’s formulae M

(m) ⊆ S

i=1

M

(m, γ

).

We now apply Lemma 2.1 to get

\

M_ε^c(m)

dx ≤ \

S_n

i=1M_ε/2^c (m,γ_i)

dx ≤ X

e

.

Since

1≤i≤n

min γ

(1 − γ

) = ε 2

 1 − ε

2



≥ ε

4 and n ≤

 2 ε (1 − ε)

 + 1 ≤

 2 ε



− 1 ≤ 2 ε we conclude

\

M_ε^c(m)

dx ≤ ne

≤ 2

ε e

≤ 2e

.

The following lemma is one of the main reasons for the exponent 2d

. Lemma 2.3 ([18], Lemma 7.2.1). Let I

, . . . , I

be subsets of {1, . . . , m}

and let e d ∈ N with P

k=1

|I

| ≥ Dm/ e d. Then X

k=1

inf n X

h∈I_k

x

: x ∈ M

(m) o

≥ Dm

2 e d

(1 − 2ε e d

).

Lemma 2.4. Let r

, . . . , r

∈ N. The number of tuples i ∈ Z

with 0 ≤ i

≤ r

(1 ≤ h ≤ m) and (i

/r

, . . . , i

/r

) 6∈ M

(m) is

r

. . . r

\

M_ε^c(m)

dx + O

 r

. . . r

min

r

 .

P r o o f. We put ξ

(x) = |{h ∈ {1, . . . , m} : x

/r

≤ γ}| and M = {x ∈ [0, r

] × . . . × [0, r

] : ξ

(x) ≤ m(γ + ε), ∀γ ∈ [0, 1]}, M

= {x ∈ [0, r

] × . . . × [0, r

] : ∃γ ∈ [0, 1] : ξ

(x) > m(γ + ε)}.

Observe that T

M^c

dx = r

. . . r

T

M_ε^c(m)

dx. We denote by G

the set of integer points of M

, thus

G

= {i ∈ Z

: (i

/r

, . . . , i

/r

) 6∈ M

(m), 0 ≤ i

≤ r

, 1 ≤ h ≤ m}.

For i ∈ Z

we put

Q

= [i

, i

+ 1] × . . . × [i

, i

+ 1].

Now we can write the assertion as

\

S

i∈GcQ_i

dx = \

M^c

dx + O

 r

. . . r

min

r



.

For x ∈ M

it follows that ([x

], . . . , [x

]) ∈ M

and hence x ∈ S

i∈G^c

Q

. In other words, M

⊆ S

i∈G^c

Q

. Therefore it suffices to show

(2.6) \

S

i∈GcQi−M^c

dx = O

 r

. . . r

min

r

 .

We have [

i∈G^c

Q

− M

= {x ∈ [0, r

] × . . . × [0, r

] :

∃γ ∈ [0, 1] : ξ

(([x

], . . . , [x

])) > m(γ + ε)} ∩ M.

Let x ∈ S

i∈G^c

Q

− M

. There exists some e γ ∈ [0, 1] with (2.7) |{h ∈ {1, . . . , m} : [x

]/r

≤ e γ}| > m(e γ + ε).

On the other hand, for all γ ∈ [0, 1] we have

(2.8) |{h ∈ {1, . . . , m} : x

/r

≤ γ}| ≤ m(γ + ε).

Observe that for all permutations π of {1, . . . , m}, (2.9) (x

, . . . , x

) ∈ [

i∈G^c

Q

− M

. Thus additionally we can assume

(2.10) x

/r

≤ . . . ≤ x

/r

and therefore we also have

(2.11) [x

]/r

≤ . . . ≤ [x

]/r

.

We put e h = |{h ∈ {1, . . . , m} : [x

]/r

≤ e γ}|. Then (2.7) and (2.11) together give

(2.12) e h > m(e γ + ε) ≥ m([x

]/r

+ ε).

If we choose γ = x

/r

, then (2.10) and (2.8) imply

(2.13) e h ≤ |{h ∈ {1, . . . , m} : x

/r

≤ x

/r

}| ≤ m(x

/r

+ ε).

The combination of (2.13) and (2.12) gives

(2.14) r

(e h/m − ε) ≤ x

< r

(e h/m − ε) + 1.

The value e h depends on x, but the possible values of e h range between 1 and m, since e h is positive. As S

i∈G^c

Q

− M

⊆ [0, r

] × . . . × [0, r

] we finally conclude from (2.14) that

\

{x∈S

i∈GcQ_i−M^c:x₁/r₁≤...≤x_m/r_m}

dx ≤ X

r

. . . r

r

= O

 r

. . . r

min

r

 .

Now (2.6) follows immediately using (2.9).

Lemma 2.5. Suppose ε > 0. Let P ∈ C[X

, . . . , X

], α ∈ C

and r ∈ N

. Let j ∈ Z

with 0 ≤ j

≤ r

(1 ≤ h ≤ m) and j

/r

+ . . . + j

/r

≤ ε.

Suppose P is M

(m)-centered at α with respect to r. Then

∂

∂X

. . . ∂X

P is M

(m)-centered at α with respect to r.

P r o o f. Since j

/r

≤ ε for all h ∈ {1, . . . , m} the lemma is an easy consequence of the definition of the set M

(m) .

2.3. Heights and Siegel’s Lemma. Let K be a number field and M (K) its set of places. Let n ∈ N. For x ∈ K

and v ∈ M (K) we put

|x|

= max{|x

|

, . . . , |x

|

} and kxk

= |x|

. If v is archimedean we put

|x|

= (|x

|

+ . . . + |x

|

)

and kxk

= |x|

. The height and the euclidean height of x ∈ K

are defined by

H(x) = Y

v∈M (K)

kxk

, H

(x) =

 Y

v∈M (K) v|∞

kxk

 Y

v∈M (K) v

-

kxk

.

We have H(x) ≤ H

(x) ≤ √

n H(x). The height of a polynomial is defined as the height of its coefficient vector. We use the notation

∆

= 1

i

! . . . i

!

∂

∂X

. . . ∂X

.

Let P ∈ Q[X

, . . . , X

] have degree ≤ r

in X

(1 ≤ h ≤ m). Let j ∈ Z

with j

≥ 0 (1 ≤ h ≤ m). We have

(2.15) H(∆

P ) ≤ 2

H(P ).

Finally, we are able to construct the auxiliary polynomial.

Lemma 2.6. Suppose 0 < ε < 1. Let F/K be an extension of number fields of degree f , let α

, . . . , α

∈ F and m ∈ N. Suppose

m ≥ 16

ε

(log(6sf ) + log ε

).

There is a constant R = R(m) such that for all r = (r

, . . . , r

) ∈ N

with r

≥ R (1 ≤ h ≤ m) there exists a non-zero polynomial P ∈ K[X

, . . . , X

] such that

(i) deg

P ≤ r

(1 ≤ h ≤ m);

(ii) P is M

(m)-centered at the points α

= (α

, . . . , α

) (1 ≤ k ≤ s)

with respect to r;

(iii) H(P ) ≤ C(F )(4H)

, where H = max{H(α

), . . . , H(α

)}

and C(F ) denotes a constant depending only on F . P r o o f. We put N = (r

+ 1) . . . (r

+ 1) and

M = |{i ∈ Z

: (i

/r

, . . . , i

/r

) 6∈ M

(m), 0 ≤ i

≤ r

, 1 ≤ h ≤ m}|.

Let P ∈ K[X

, . . . , X

] with (i). We need to determine the coefficients of P such that (ii) and (iii) hold. (ii) says that

(2.16) ∆

P (α

) = 0

for all i ∈ Z

with (i

/r

, . . . , i

/r

) 6∈ M

(m), 0 ≤ i

≤ r

, 1 ≤ h ≤ m and 1 ≤ k ≤ s. (2.16) is a system of linear equations, where the unknowns are the coefficients of P . To solve (2.16) we will apply Siegel’s Lemma in the form given by Bombieri and Vaaler [2].

Lemma 2.4 says

M = r

. . . r

\

M_ε^c(m)

dx + O

 r

. . . r

min

r



and therefore it follows from Lemma 2.2 that M

N = \

M_ε^c(m)

dx + 1

(r

+ 1) . . . (r

+ 1) · O

 r

. . . r

min

r



< 2e

+ 1

(r

+ 1) . . . (r

+ 1) · O

 r

. . . r

min

r

 . Hence for large r

, . . . , r

we get

(2.17) M/N < 3e

. By assumption,

m ≥ 16

ε

(log(6sf ) + log ε

) = 16

ε

log(6sf /ε).

This is equivalent to

(2.18) 3sf e

≤ 1/2.

The inequalities (2.17) and (2.18) together give

(2.19) sf M < N sf 3e

≤ N/2.

If we denote by A the matrix corresponding to (2.16), then by [2], Theorem 12 and (2.19) we get a non-zero polynomial P ∈ K[X

, . . . , X

] satisfying (i), (2.16) and

H(P ) ≤ C(F )( max

a row of A

H

(a))

≤ C(F ) max

a row of A

H

(a),

where C(F ) denotes a constant only depending on F . By standard estimates

we know that H

(a) ≤ (4H)

and the lemma follows.

3. Roth’s Lemma. The essential ingredient to Roth’s Theorem in [13]

is the so-called Roth’s Lemma. We quote its version proved by J. H. Evertse [6], which is a quantitative improvement on the original. J. H. Evertse proved this result by using Faltings’ Product Theorem [9].

Let P be a non-zero polynomial in unknowns X

, . . . , X

with complex coefficients. Let α ∈ C

and r ∈ N

. We define, as in [13],

Ind

P

= min{i

/r

+ . . . + i

/r

: ∆

P (α) 6= 0, i ∈ Z

, i

≥ 0, 1 ≤ h ≤ m}

and say that P has index Ind

P at α with respect to r.

Proposition 3.1 ([6], Theorem 3). Let m be an integer ≥ 2, let r = (r

, . . . , r

) be a tuple of positive integers, let P ∈ Q[X

, . . . , X

] be a non- zero polynomial of degree ≤ r

in X

for h = 1, . . . , m and let 0 < ε ≤ m+1 be such that

(3.1) r

/r

≥ 2m

/ε for h = 1, . . . , m − 1.

Further , let β

, . . . , β

be algebraic numbers with

(3.2) H

((1, β

))

> {e

H

(P )}

(1 ≤ h ≤ m).

Then Ind

P < ε.

4. A quantitative result. Suppose 0 < ε < 1. Let F/K be an extension of number fields of degree f . Let S be a finite subset of M (K) of cardinality s. Suppose that for each v ∈ S we are given fixed elements α

∈ F . Suppose H(α

) ≤ H (v ∈ S). Let m ∈ N with m ≥ (16/ε

)(log(6sf ) + log ε

).

Under these assumptions the hypotheses of Lemma 2.6 are satisfied.

Let R = R(m) be the constant given by Lemma 2.6. Suppose r

≥ R (1 ≤ h ≤ m). Then there is a polynomial P with

(4.1) P ∈ K[X

, . . . , X

], P 6= 0;

(4.2) deg

P ≤ r

(1 ≤ h ≤ m);

(4.3) P is M

(m)-centered with respect to r

at the points α

= (α

, . . . , α

) (v ∈ S);

(4.4) H(P ) ≤ C(4H)

, where C = C(F ) is a constant just depending on F .

Lemma 4.1. Suppose 0 < δ ≤ 1, d ∈ N and 0 < ε ≤ δ/(20d

). Let Γ be a tuple of non-negative integers with

X

v∈S

Γ

= 1 − δ/(24d

).

Suppose there are elements β

, . . . , β

∈ Q satisfying [K(β

) : K] ≤ d,

H(β

)

≤ H(β

)

≤ H(β

)

(1 ≤ h ≤ m), (4.5)

kα

− β

k

< H(β

)

(1 ≤ h ≤ m, v ∈ S) (4.6)

and

(4.7) H(β

)

≥ max{C

, 2

H

} (1 ≤ h ≤ m).

Then Ind

P > ε.

P r o o f. Let j ∈ Z

with 0 ≤ j

≤ r

, 1 ≤ h ≤ m and (4.8) j

/r

+ . . . + j

/r

≤ ε.

Put

(4.9) T (X) = X

i

a

X

. . . X

= ∆

P (X).

We have to show

(4.10) T (β) = 0.

First we establish an inequality for the height of T . From (2.15), (4.2), (4.4) and (4.7) we get

(16H

)

H(T ) ≤ (2

H

)

H(P ) ≤ C(2

H

)

≤ C

 Y

h=1

H(β

)



.

By (4.7) we have C ≤ H(β

)

≤ Q

h=1

H(β

)

and therefore (4.11) (16H

)

H(T ) ≤

 Y

h=1

H(β

)



.

We will need (4.11) later on.

Put E = K(β

, . . . , β

). We denote by E ,→ K

the set of K-embeddings of E into K

, i.e. the homomorphisms of E in K

which are the identity on K.

For each place w of E which lies over v of K, there exists a λ ∈ E ,→ K

with

|a|

= |λ(a)|

for all a ∈ E. There are in fact [E

: K

] such embeddings.

With these notations the product formula reads Y

p∈M (Q)

Y

w∈M (E) w|p

|x|

= Y

p∈M (Q)

Y

v∈M (K) v|p

Y

λ∈E,→K^K v

|λ(x)|

= 1

for all x ∈ E

. From (4.1) and (4.9) we know that T has coefficients in K and hence T (β) ∈ E. Therefore to prove (4.10) it suffices to show

(4.12) Y

p∈M (Q)

Y

v∈M (K) v|p

Y

λ∈E,→K^K v

|T (λ(β))|

< 1.

Let v ∈ M (K). In the sequel we estimate Q

λ∈E,→K^K _v

|T (λ(β))|

. Put

(4.13) κ

=

 1 if v | ∞, 0 if v - ∞

and r = r

+ . . . + r

. For v 6∈ S, by trivial estimates of (4.9) we get Y

λ∈E,→K^K _v

|T (λ(β))|

≤ Y

λ∈E,→K^K _v



2

max

i

|a

|

Y

|(1, λ(β

))|

 (4.14)

= 2

max

i

|a

|

Y

λ∈E,→K^K v

1≤h≤m

|(1, λ(β

))|

.

Now, let v ∈ S. We can write (4.6) as (4.15) |µ

(α

) − µ

(β

)|

< H(β

)

(1 ≤ h ≤ m, v ∈ S), where µ

denotes a fixed K-embedding of Q in K

. Let λ ∈ E ,→ K

. We expand T (X) around the point µ

(α

) = (µ

(α

), . . . , µ

(α

)) in a Taylor series to get

(4.16) |T (λ(β))|

≤ 2

max

i

∆

T (µ

(α

)) Y

(λ(β

) − µ

(α

))

. By trivial estimates we get

(4.17) |∆

T (µ

(α

))|

≤ 4

max

ei

|a

|

|(1, µ

(α

))|

. The main term we have to look at is max

Q

h=1

|λ(β

) − µ

(α

)|

, where the maximum is taken over all i with ∆

T (µ

(α

))

6= 0. By (4.3), (4.8), Lemma 2.5 and µ

(∆

T (α

)) = ∆

T (µ

(α

)), for tuples i satisfy- ing ∆

T (µ

(α

))

6= 0 we have (i

/r

, . . . , i

/r

) ∈ M

(m). Therefore it suffices to consider the term sup

Q

h=1

|λ(β

) − µ

(α

)|

. If λ(β

) = µ

(β

), we can estimate the factor satisfying (4.15) non- trivially. Hence we treat the cases λ(β

) = µ

(β

) and λ(β

) 6= µ

(β

) separately. Put

I

= {h ∈ {1, . . . , m} : λ(β

) = µ

(β

)}.

We have

(4.18) sup

x∈M2ε(m)

Y

|λ(β

) − µ

(α

)|

≤

 sup

x∈M_2ε(m)

Y

h6∈I_λ

|λ(β

) − µ

(α

)|



×

 sup

x∈M2ε(m)

Y

h∈Iλ

|λ(β

) − µ

(α

)|

 . We estimate the first factor of (4.18) trivially and get

(4.19) sup

x∈M_2ε(m)

Y

h6∈I_λ

|λ(β

) − µ

(α

)|

≤ Y

2

|(1, λ(β

))|

|(1, µ

(α

))|

. For the second factor of (4.18) we use (4.15) and (4.5) to get

sup

x∈M2ε(m)

Y

h∈I_λ

|λ(β

) − µ

(α

)|

= sup

x∈M_2ε(m)

Y

h∈Iλ

|µ

(α

) − µ

(β

)|

< sup

x∈M_2ε(m)

Y

h∈Iλ

H(β

)

≤ sup

x∈M_2ε(m)

H(β

)

P

h∈Iλxh

= H(β

)

P

h∈Iλx_h

. Taking the product over all K-embeddings of E into K

gives

(4.20) Y

λ∈E,→K^K _v

sup

x∈M_2ε(m)

Y

h∈I_λ

|λ(β

) − µ

(α

)|

< H(β

)

2+δ)_{[Kv :Qp]}^[K:Q] P

λ∈EK ,→Kv

inf_x∈M2ε(m)P

h∈Iλx_h

. To apply Lemma 2.3 we need a lower bound for P

λ∈E,→K^K v

|I

|. Let δ

denote the Kronecker symbol. We have X

λ∈E,→K^K v

|I

| = X

λ∈E,→K^K v

|{h ∈ {1, . . . , m} : λ(β

) = µ

(β

)}|

= X

λ∈E,→K^K v

X

1≤h≤m

δ

= X

1≤h≤m

X

λ∈E,→K^K v

δ

= X

1≤h≤m

[E : K(β

)] ≥ X

1≤h≤m

[E : K]

d = m[E : K]

d .

Now we apply Lemma 2.3 to (4.20) with [E : K] in place of D and 2ε in place of ε to get

(4.21) Y

λ∈E,→K^K _v

sup

x∈M_2ε(m)

Y

h∈Iλ

|λ(β

) − µ

(α

)|

< H(β

)

. The combination of (4.18), (4.19) and (4.21) gives

(4.22) Y

λ∈E,→K^K v

sup

x∈M2ε(m)

Y

|λ(β

) − µ

(α

)|

<

 Y

λ∈E,→K^K _v 1≤h≤m

2

|(1, µ

(α

))|

|(1, λ(β

))|



× H(β

)

< (2

|(1, µ

(α

))|

)

 Y

λ∈E,→K^K v

1≤h≤m

|(1, λ(β

))|



× H(β

)

and the combination of (4.16), (4.17) and (4.22) gives

(4.23) Y

λ∈E,→K^K v

|T (λ(β))|

< Y

λ∈E,→K^K v

2

max

i

|∆

T (µ

(α

))|

Y

|λ(β

) − µ

(α

)|

≤ 8

max

ei

|a

|

|(1, µ

(α

))|

× Y

λ∈E,→K^K _v

max

i

Y

|λ(β

) − µ

(α

)|

< 16

max

i

|a

|

|(1, µ

(α

))|

×

 Y

λ∈E,→K^K v

1≤h≤m

|(1, λ(β

))|



H(β

)

.

Finally, if we take the product over all valuations of E, then (4.14) and

(4.23) together lead to

(4.24) Y

p∈M (Q)

Y

v∈M (K) v|p

 Y

λ∈E,→K^K _v

|T (λ(β))|



<  Y

p∈M (Q)



16

Y

v∈S, v|p

|(1, µ

(α

))|



×  Y

p∈M (Q)

 Y

v∈M (K) v|p

max

|a

|





×  Y

p∈M (Q)

Y

v∈M (K) v|p

 Y

λ∈E,→K^K v

1≤h≤m

|(1, λ(β

))|





× H(β

)

For the middle term of the right-hand side of (4.24) we have (4.25)

 Y

p∈M (Q) v∈M (K),v|p

max

|a

|



× Y

p∈M (Q) v∈M (K),v|p

 Y

λ∈E,→K^K _v 1≤h≤m

|(1, λ(β

))|



=

 Y

p∈M (Q) v∈M (K),v|p

max

|a

|



× Y

p∈M (Q) v∈M (K),v|p

 Y

w∈M (E),w|v 1≤h≤m

|(1, β

)|



=

 Y

v∈M (K)

max

ka

k



 Y

h=1

Y

w∈M (E)

k(1, β

)k



= H(T )

 Y

h=1

H(β

)



.

Before we estimate the first term of the right-hand side of (4.24) we make some remarks: For each v ∈ S there exists some w

∈ M (F ) such that

|µ

(x)|

= |x|

for all x ∈ F , hence |(1, µ

(α

))|

≤ H(α

)

. Further from (4.13) we have

Y

p∈M (Q)

16

= 16

.